Expansion—contraction behaviour of non-Newtonian jets

Chemical Engmeermg Science, 1968, Vol 23, pp 1165-l 172

Expansion-contraction

Pergamon Press

Prmted m Great Bntam

behaviour of non-Newtonian jets

W KOZICKI and C TIU Department of Chemical Engmeermg, University of Ottawa, Ottawa 2, Canada (Fwst recerued 2 1 March 1968, m reulsedform 2 1 May 1968)

Abstract-An analysts of the expansion-contraction behavlour of lammar viscoelastlc non-Newtoman Jets IS presented which extends previous analyses by conslderatlon of the anomalous behavlour, as characterized by an effective velocity of shp, exhibited by the fluid at the solid-fluid Interface An apparent improvement m agreement ISrealized m the first devlatonc normal stress ddference (PI1- P2J obtamed m the high shear rate remon by the extrusion method with values obtamed by near linear extrapolation of rheogomometer data for the low shear rate reDon when this allowance for the anoma.lous behavlour ISmade m the evaluation of the normal stress difference INTRODUCTION

Jets have been the subJect of considerable experimental and theoretlcal mvestlgatlon of late, prompted by practical problems encountered m processmg and manufacturing mdustnes such as the filament industry, and by theoretical conslderatrons as to the rheologlcal behavlour exhlblted by the fluids when extruded The mvestlgatlons of Pearson[l] mto the meltflow mstablhty of extruded polymers serve to exemplify a problem of practical and theoretical interest Much of the past work relatmg to Jet studies has been reported by Mlddleman[2] and Metzner et al [3-61 and IS not repeated here Gaskms and Phlhppoff [7] suggested measurement of the expansion of a Jet of fluid issuing from a capillary (the Barus or Merrmgton effect) as a convenient means of measurmg normal stresses at high shear rates Investlgatlons mto this area were also conducted by Gill and Gavls [8], Sakladls[9], Phlhppoff and Gaskms[lO], and Metzner et al. [3] Subsequently, Harns[ 11, 121 pointed out that the measurement of the Jet diameter introduced an unnecessary artifice, since the thrust of theJet 1s the pnmary quantity required for determmatlon of the normal stress Metzner and colleagues [3-61 developed analyses for use m the evaluation of the normal stress from measurements of either the Jet diameter or the thrust of the Jet and conducted extensive expenmental measurements and evaluations of LIQUID

normal stresses m support of the analyses presented The purpose of this mvestlgatlon 1s to extend, m a single analysis, the apphcablhty of the prevlous analyses of Metzner and colleagues[l3] to physical sltuatlons m which anomalous behavlour at the solid-fluid interface, as charactenzed by an effective velocity of slip at the sohd surface, 1s encountered The present analysis applies equally to data avadable m the form of either Jet diameter or thrust measurements It IS shown that an apparent improvement IS reahzable m the concordance between extrapolations of rheogomometer determinations of the first devlatonc normal stress difference and values determined by the Jet method If the anomalous behavrour m the wall region of the die IS accommodated m the analysis of the data ANALYSIS

The assumptions made concerning the physical problem in the previous developments of Metzrier and colleagues[3,4,6,14] are also made m the present analysis with the exceptlon that allowance IS Bven for the anomalous behavlour of the fluid The vahdlty of the assumptions involved has been dlscussed m great detad by these authors The relattonshlp obtained when the net thrust of the fluid lssumg from a tube IS equated to the difference between the momentum flux of the

1165

tt

5-A

W

KOZICKI

and C TIU

fluid leavmg the tube and the tenstle stresses exerted on the flurd at the exrt sectton of the tube

Tw= i-4(u- d2) - (~11)

(7)

where T, ISgiven by

1s

T = j.I pu2dA -j-j-

rndA.

T,=T/A-22pu,((u)-~~)-ppu,~

(1)

Here, 7r1 denotes the normal component of the total stress tensor m the axial du-ectton The total stress tensor, r, 1s related to the tsotroptc pressure and to the devtatorrc tensor by the equation ?T=--pI+P

(2)

where I 1s the umt tensor, P 1s the devratortc tensor andp IS the rsotroprc pressure given by P=-4fm=-~(7*1+722+733)

TM = p(A/A,) (u)”

which IS the analogous expression to one denved and utrhzed previously [ 141 without constderatron of the anomalous wall effect Taking the partial denvattve of Eq (9) with respect to r, (at a grven tube dmmeter) gives (q,,,=$&T,(l+;E-j

(10)

where I’ has been used to denote the integral I'=

,:(u-

U,)2rdT

(11)

The remamder of the analysis for (rll)W 1s substantrally the same as presented elsewhere [ 141, recognizmg that m the present case,

(4)

and a single analysis of the extrusron problem, m terms of T/A as suggested by Harns [ 11, 121, suffices also for the descrlptton of the problem expressed m terms of the relatIveJet srze Equation (1) may be rewritten as follows TM = P(u”) - (711)

For a cu-cularlet, Eq (7) leads to

(3)

The thrust of the Jet T IS amenable to evaluation by direct measurement[6, 121 or can be established from a knowledge of the relative size of the Jet using the expresston for the thrust of the Jet, T = A,puj2, combined with the equatton of contmuny, Apu, = Ap(u) The specrfic thrust 1s thus determined readrly m the latter case by the simple relatronshrp

(8)

u--u,=:r&)dr,

(12) Or

8((u)

-‘to)

D

=

4

@J 72f(T)

&,

ruJ I 0

(13)

and n’ =

(5)

dlnrW ,lnW)-&) D

The effective velocrty of slip at the wall, u,, charactenzmg anomalous behavrour along the crrcumference of the extrusron tube, IS mtroduced by expansron of the first term on the rtghthand side of Eq (5) according to P(U”> = p[((u---XJ2)

+2&0((u)--u,)

from whtch rt also follows that

+4n21 (6)

Thus, Eq (5) becomes, after some rearrangement,

f(Tw)=

1+3n’ 4n,

8((U)-Uu,) D [

1

(15)

The followmg result analogous to the expres1166

Expansion-contraction behavlour of non-NewtonIanJets

ston of Shertzer obtamed

and

Metzner [ 141 1s finally

whrch can be expanded to grve A,=A:(l--s)~+2s--s$$

(21)

(T11)w= P((U> -Q[~

alnT,

l+$

-T,

In arnvmg at Eq (2 I). (u”) m the first term on the nght side of Eq (20) 1s expanded accordmg to Eq. (6), and (u)’ m the denommator of the same term IS replaced by

alng((~)

L

-u,) D

(16) 1

(u)2

=

((4

-kJ2

(22)

(1 -s)2 or, uttlizmg Eq (14) and mtroductlon defined by Eq (23),

of A’, where s = u,/(u) , and,4 k IS grven by A,

r (17) The quantrty (3n’+ 1)/n’ IS found, wtth the help of Eq (14), to be gtven by the followmg expression* 3n’+l n’

_

Tw%Tw)

(23)

((u)-u)2

The quantrty A :, also Introduced m Eq (I 7), IS the value of A, for a flmd not exhrbrtmg anomalous behavrour at a solid boundary and wtthout normal stresses, as seen m Es (2 1) The foliowmg expresstons for the quantny Ai are found wtth the help of Eqs (12) and ( 13) 27,’

I ‘?f(T)dT

Jb’”

TdT[ I,“f(T)

dT]’

A!= [ lTuTy.(T)

For an Elhs flurd, f(r)

=((-d2)

= (T/W,) [I + (T/TI/~)~-‘I 7

=

the quantity (3n’+ 1)/n’ yields

~,?f(d dT j-+(T)dT

27a2

(24) [l;T’f(T)dT]’

Equatrons (4) and (5) can be combmed to obtam the ratro of the cross-sectronal area of theJet to that of the IssuingJet,

dT]’

-

The derrvatlon of the latter result. whrch IS m a more convement form, m the above equatron from the former expressron mvolves successive Interchanges m the order of mtegratlon whtch are mdrcated m the appendrx The form of the expressron assumed by AL for the Ellis fluid IS

I

(25) A,-

A _ (If’)

(711)

A,

Pi

(u)”

(20) whrch yrelus the expected

value of 4/3 for a Newtoman flurd on settmg a! equal to umty

1167

W KOZICKI RESULTS

OF ANALYSIS

Ftgures 1 and 2 present the expertmental data of Shertzer [ 151 plotted m the standard manner as rw vs 8(u)/D or 8((u) -u,)/D for 0.5% J-100 m water and 3% PIB m decahn, respecttvely In the TVvs 8( u)/D plots separate curves are drawn through the pomts representing different capillary tube diameters, consistent wtth anomalous behaviour at the solid boundary When the same data are plotted as 8((u) - u,)/D vs rw, the effective velocity at the wall U, estabhshed

K

OA.05422cm A 6, 02666cm 0

0,

01146

v

E,

00636cm

cm

WJVD or 8((u) - u&D , set-’ Fig I Flow curves determmed from Shertzer’s data for 0 5% J-100 m water solution (open pomts-T, vs 8(u)/D, solldpomts--7,vs 8((u) --u,)/D)

0

02666

cm

A

01146

cm

0

o.o63!lcm

t

EM/D

or

8((u>-u,~)/D,

1

a.2

Fig 2 Flow curves determmed from Shertzer’s data for 3% PIB m decalm solution (open pomts--7, vs 8(u)/D, sohd pomts--7w~s 8((u)-u,)/D)

and C TIU

from the separation of the former mdtvtdual curves by the standard method[l6], the points merge so as to be sattsfactonly represented by smgle curves (uppermost curve m each figure wtth sohd points), as expected. The latter curves were used m evaluatton of the shear stressshear rate relationshtp charactenzmg the ~LSXM.I~ flow behavtour of the fluids Because of the curvature m the curves, the flow behavtour index n’, determined as the slope of the tangent to the curve at a point, 1s not constant but vartes wtth the wall shear stress from 0.39 to O-856 for the J-100 solution and from 0.43 to 0.84 for the PIB solutton Figure 3 shows a plot of the effective velocity at the wall U, vs wall shear stress T,,,for both the J-100 and PIB soluttons, computed from the cap&try tube data Postttve effective shp veloctties were obtained m both cases suggestmg separatton of the solute nolvmer molecules from the solvent at the sohd boundary, the socalled “separation phenomenon” as the probable mechanism at the sohd-fluid interface The effective velocity of slip u, is satisfactorily represented by a linear function of the shear stress at the wall TV The slip coeffictents 5 computed are 0 694 and 0 187 ftYlb,-set for the J-100 and PIB soluttons, respecttvely Figures 4 and 5 are plots of the deviatonc normal stress dtfference (PI1 - P22)u, vs shear rate I for the 0.5% J-100 m water and 3% PIB m decahn soluttons, respectively, computed from the thrust measurements of Shertzer on these fluids by the present method taking into constderatlon the effect of anomalous behavtour The broken lines m each figure represent the previous correlattons and extrapolations The shaded area and the pomts m the lower left-hand corner of Figs 4 and 5, respectrvely, represent data obtained from rheogomometer measurements, plotted as gtven by Shertzer The pomts plotted m Fig. 4 represent values of (PI1 -P&, yielded by the present method of analysts (Eq 16), which are generally lower than the values determmed without consideration of anomalous behavtour It IS seen that the mterpolatton hne connecting the rheogomometer and

1168

Expansion-contractlon behavlour of non-Newtomaqets

PSf t Fig 3 Plot of the effective velocity of shp vs wall shear stress I

VMOO~NI~METER

08m

4

L

r ,

m-1

Fig 4 Devlatonc normal stress chfference vs shear rate from Shertzer’s thrust measurements on 0 5% J-100 solution data for the J-100 solution m Fig. 4 has less curvature (almost hnear) than the one obtamed by Shertzer This suggests the possiblhty of predctlon, to quite a good degree of approxlmatlon, of values of the normal stress difference m the high shear-rate reDon by simple extrapolatlon of rheogomometer data obtamed in the thrust

low shear-rate regon, m contrast with previous expenence A slmdar extrapolation can also be achieved with the measurements for the PIB solution (Fig 5) However, the Qfferences between the previous values of (PI1 -P&,, and those determmed by the present analysis for the PIB solutron are not so large as m the case

1169

W KOZICKI

8

RHEO6ONlOMETER

0

02666

and C TIU

DATA

cm

v

02222cm

A

01146

cm

D

00635

cm

&Y

a?

05422

A

02666

cm

0

0

01146

cm

0

v

Q0835 cm

A

IO’

I d

r ,

cm

0

l

, IO4

_

I

4 Id

8(d/D or 8((u)-u,,VD, sac-’ Fig 6 Dewatonc normal stress dlfference vs 8(u)/D 8((u) - u,)/D curves for 0 5% J-100 solution

SW-’

Fig 5 Dewatonc normal stress dlfference vs shear-rate from Shertzer’s thrust measurements on 3% PIB solutmn

of the J-100 solution, due to the lower effective shp velocittes found for the PIB solutton The scattering of the data points m Figs 4 and 5 are attributable to two major sources of error (1) a certain lack of precision m the expertmental measurements, (2) uncertamties existmg m the T, vs 8((u) - u,)/D correlation Aberrations m the latter are magnified because of the form of Eq (16), which mvolves taking a difference, which is small, between two large terms m the evaluatton of (P,,-Pp,& A slight alteration m the slope determmed from the log T, vs log 8((u) - u,) /D plot affects the final result agmficantly Shertzer has emphasized m discussion of the thrust measurements that a small error m the thrust T could magmfy the error m (PI, - P22)Wby several orders Figure 6 IS a plot showmg the normal stress difference plotted agamst 8(u)/D and 8((u) u,)/D for 0.5% J-100 m water, which is an analogous plot to Figs. 1 and 2. By analogy with these figures, the plots of (PI1 -P2& agamst 8(u)/D also yield dtstmct curves, characteristic of the capillary diameter, deptctmg the anomalous surface effect Addittonally, the upper curve represents all of the data obtained wtth

-

and

&fferent capdlanes plotted as (PI1 -P2.& vs 8((u) - u,)/D, which are now superimposed on a single smooth curve, except for some scattering of data points m the lower shear-rate region This scattenng is attributable to the experimental uncertainty m the thrust measurement using Tube A, as ongmally reported by Shertzer. This figure serves to illustrate the effect of the existence of anomalous surface behavtour on the normal stress measurement plotted m this manner A similar plot obtained for the PIB solution is not shown here Figure 7 shows values of (PI1 -I’.&, calculated for the same J-100 and PIB solutions from Jet diameter measurements taken by Shertzer using a photographtc technique These were computed taking mto consideration the anomalous surface effect using Eqs. (16) or (17), and plotted agamst shear-rate as shown The data are correlated successfully by a straight line on a logarithmic plot over the relatively narrow shear rate range The new values of the normal stress difference are somewhat lower than those determined by Shertzer SUMMARY

A generalization of the lammar flow behavtour of non-Newtonian Jets has been presented which

1170

Expansion-contractlon

behavlour of non-NewtonIanJets

IO3I I’ : _ 3

74 cl? ‘- Id-

ri

J n’

-

_

.II,,

A

01146

0

00635cm

IO'

cm

Id

r , see-'

s

Fig 7 Devlatonc normal stress tierence vs shear-rate determmed for 0 5% J-100 and 3% PIB solutions from let diameter measurements

T

T, accounts for anomalous behavtour at the sohdfluid interface wrthm the tube An improvement m agreement m the devtatonc normal stress dtfference determmed by the extrusion method with extrapolated rheogomometer data was achieved by taking mto constderatron the anomalous behavtour of the fluid Acknowledgment -The authors wish to gratefully acknowledge the financial assistance received from the Natlonal Research Councd of Canada The second author was also the reclplent of a Natlonal Research Councd StudentshIp

NOTATION

A A, A,

area of flow cross s&ton area of the Jet stream area ratio defined by Eq (20)

function defined by Eq (23) diameter of the captllary tube function of shear stress denoting velocity gradient, charactertsttc of flmd unit tensor integral defined by Eq (11) (subscript) refers to theJet flow behavtour index defined by Eq (14) isotropic pressure devtatonc stress tensor, PII denotes the component m the du-ectlon of fluid flow, Pz2 denotes the component m the dtrectton normal to the plane of shear, and Ps3 denotes the component m the neutral du-ectton = U,/(U), ratio of the effective shp velocity to the average velocity thrust of the fluid tssumg from a tube function defined by Eq (8) local velocrty average velocity effective velocity of shp at the wall (subscript) refers to condtttons extstmg at the capillary wall dummy vat-tables

(u”, uw W

xvY

Greek letters a, 770,~1/2 = parameters m the Ellis fluid model denstty of the fluid P 7 total stress tensor shear stress average normal stress m the drrectton (71:) of the fluid flow l- shear rate 5 = u,/r,, slip coefficient

REFERENCES J R A and PETRIE C J S , Proc 4th Inr Congr Rheol 1965, p 265 PI MIDDLEMAN S , Ind Engng Chem Fundls 1964 3 I 19 W T , SAILOR R A and WHITE J L , Trans Sot Rheol 19615 133 131 METZNER A B , HOUGHTON W T, HURD R E and WOLFE C C , IUTAM Int Symp on Second r41 METZNER A B , HOUGHTON Effects rn Elastrc~ty, Piastlclty and Fluid Dynamrcs, Halfa, 1962, p 650 r51 GINN R F and METZNER A B , Proc 4th Inr Gong Rheol 1965, p 583 [61 SHERTZER C R and METZNER A B , Proc 4th Int Cong Rheol 1965, p 603 HI PEARSON

1171

Order

W KOZICKI

and C TIU

[7] GASKINS F H and PHILIPPOFF W , Tram Sot Rheol 1959 3 181 [8] GILLS J andGAVIS J ,J Polym Scr 195620287 [9] SAKIADIS B L A I Ch E Jf 1962 8 3 17 [IO] PHILIPPOFF W and GASKINS F H , Trans Sot Rheol 1958 2 263 [ 1 I] HARRIS J , Nature 1961 190 993 [ 121 HARRIS J , Proc 4th Int Cong Rheof 1965, p 417 [13] WHITE J L and METZNER k B , Truns Sot Rheol 19637 295 1141 SHERTZER C R and METZNER A B . Tram Plust Inst Land 1963 31148 ii5j SHERTZER C R , M SC Ch E Thesis, Uklverslty of Delaware I964 [ 161 KOZICKI W , CHOU C H and TIU, C , Chem Engng Scr 1966 21665

APPENDIX Development of thejinal relatron for AL In this sectlon, it IS shown that the second hne of Eq (24) follows directly from the first hne An mtroductlon of a change m dummy variables and an interchange m the order of mtegratlon of the resulting triple integral are effected as follows

A second interchange m the order of mtegratlon of Eq (I A), and replacement of the dummy vanables x and y by 7 yields

634) The above result when combined with the first hne of Eq. (24) yields the desired expression gven by the second line

R&urn&- Une analyse du comportement expansion-contra&on de Jets non-newtomens lammrures vlsc&lastlques, est prksent6e Celle-a prolonge des analyses anttneures en consld&ant le comportement anomal, amsl qu’d est caract&& par une velocrtt effective de ghssement que montre le flulde B I’mterface sohde-flulde Une amehoratlon apparente de I’accord est rkahsee dans la premltre d&rence normale de contramte dCvlatonque (PII-P& obtenue dans la zone de grande separation par la methode de I’extruslon avec des valeurs obtenues par extrapolation hnCalre proche des don&es rhtogomometnques pour la zone de falble stparatlon quand on permet le comportement anomal en tvaluant la ddT&ence de la contramte normale Zusammenfassung - Eme Analyse des Expanslons/Kontraktlons-Verhaltens von lammaren vlskoelastischen, mchtnewtonschen Strahlen wlrd erortert Sle erweltert fruhere Analysen durch Beruckslchtlgung des anomalen Verhaltens, das durch eme wlrksame Schlupfgeschwmdlgkelt der Flusslgkelt an der Grenzflache zwlschen Feststoff und Flusslgkelt gekennzelchnet 1st Es wlrd eme schembar bessere Uberemstlmmung fur die erste Abwelchung der normalen Spannungsdlfferenz (P,, - P,,) erzlelt, die man m der Zone mlt hoher Schergeschwmdlgkelt durch die Extruslonsmethode mlt werten unter Anwendung nahezu hnearer Extrapolation der Rheogomometerdaten fur den Berelch der medngen Schergeschwmdlgkelt gewmnt, wenn dlese Abwelchung, die durch das anomale Verhalten bedmgt Ist, be1 der Beurtedung der normalen Spannungs dlfferenz beruckslchtlgt wlrd

1172